Apparatus and method for correcting phase error

ABSTRACT

An apparatus for correcting a phase error is provided. The apparatus includes an error estimating module and a correcting module. The error estimating module receives a phase-shift keying signal, and calculates a phase error according to the phase-shift keying signal, a plurality of known candidate signals and Bayesian estimation. The correcting module corrects the phase-shift keying signal according to the phase error.

This application claims the benefit of Taiwan application Serial No.101114608, filed Apr. 24, 2012, the subject matter of which isincorporated herein by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates in general to a digital communication technique,and more particularly, to a technique for determining a phase error of acommunication signal.

2. Description of the Related Art

Phase-shift keying (PSK) is a modulation scheme that conveys data bychanging or modulating a phase of a signal. Phase-shift keying isprevalent in the digital communication field as it has high datatransmission efficiency. Based on the number of candidate phases in aconstellation, phase-shift keying is further categorized into binaryphase-shift keying (BPSK), quadrature phase-shift keying (QPSK) and8-PSK, etc.

For a phase-shift keying communication system, an important task of areceiver is to determine a phase of a currently received signal. Thereceiver can only retrieve information carried by the received signalafter the phase of the received signal is accurately determined.However, in a wireless communication system, interference may be presentdue to factors such as channel noise and circuit mismatching in thereceiver; thus, accurately determining the signal phase has become agreat challenge.

SUMMARY OF THE INVENTION

To achieve the above goal, the invention is directed to an apparatus andmethod for correcting a phase error, in which Bayesian estimation isutilized for minimizing an error between an estimated phase result and acorrect phase. Since Bayesian estimation is capable of providingoptimized maximum a posteriori (MAP) performance in an additive whiteGaussian noise (AWGN) channel, the phase error correcting apparatus andmethod can accordingly generate a satisfactory phase estimation result.In addition, by selectively decreasing the number of candidate phases,costs and complexities of the phase error correcting apparatus andmethod can be further reduced.

A phase error correcting apparatus is provided according to oneembodiment of the present invention. The phase error correctingapparatus comprises an error estimating module and a correcting module.The error estimating module receives a phase-shift keying signal, andcalculates a phase error according to the phase-shift keying signal, aplurality of known candidate signals and Bayesian estimation. Thecorrecting module corrects the phase-shift keying signal according tothe phase error.

A phase error correcting method is provided according to anotherembodiment of the present invention. The method comprises steps of:receiving a phase-shift keying signal; calculating a phase erroraccording to the phase-shift keying signal, a plurality of knowncandidate signals and Bayesian estimation; and correcting thephase-keying shift signal according to the phase error.

In practice, the apparatus and method for correcting a phase shift canbe implemented in not only phase-shift keying digital communicationsystems but also other types of signal processing systems with a demandof determining a signal phase error, so as to provide satisfactory phaseerror determination capability.

The above and other aspects of the invention will become betterunderstood with regard to the following detailed description of thepreferred but non-limiting embodiments. The following description ismade with reference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a phase error correcting apparatus according to one embodimentof the present invention.

FIGS. 2A and 2B are block diagrams of a phase error correcting apparatusaccording to other embodiments of the present invention.

FIGS. 3A and 3B are block diagrams of a phase error correcting apparatusfurther comprising a selecting module according to other embodiments ofthe present invention.

FIG. 4 is an example of an 8-PSK phase constellation.

FIG. 5 is a flowchart of a phase error correcting method according toone embodiment of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 shows a phase error correcting apparatus 100 according to oneembodiment of the present invention. The phase error correctingapparatus 100 comprises an error estimating module 12 and a correctingmodule 14. In practice, the phase error correcting apparatus 100 may beintegrated in various types of digital communication systems adoptingphase-shift keying (e.g., DVB-S2 digital television broadcastingsystems) or other signal processing systems with a demand of determininga signal phase error. In the description below, a phase-shift keyingsignal is taken as an example of a signal received by the phase errorcorrecting apparatus 100.

Assuming an original noise-free phase-shift keying signal transmittedfrom a transmitter is a, a phase-shift keying signal x received by thephase error correcting apparatus 10 is then a sum of the originalphase-shift keying signal a and Gaussian noise n: x=a+n. A differencebetween an estimation result ‘â’ generated for the original phase-shiftkeying signal ‘a’ by the phase error correcting apparatus 100 and theoriginal phase-shift keying signal a is denoted as ε. In thisembodiment, the phase error correcting apparatus 100 adopts the squareof the difference ε as a quadratic cost function C(ε), and utilizes anexpected value of the function C(ε) as a Bayesian risk R:R=E[C(ε)]=E[ε ² ]=E[(â−a)²]

The original signal a is a phase-shift keying signal, meaning that it isone of a plurality of known candidate signals s_(i). Taking quadraturephase-shift keying (QPSK) as an example, the number of candidate signalss_(i) is four, which are respectively 45-degree, 135-degree, 225-degreeand 315-degree signals. The candidate signals s_(i) are also known tothe phase error correcting apparatus 100. A main goal of the phase errorcorrecting apparatus 100 is to identify the estimation result ‘â’corresponding to ‘a’ minimized Bayesian risk R from the candidatesignals s_(i), that is, to identify the estimation result â having theminimum error ε, so that the estimation result a can be closest to theoriginal phase-shift keying signal ‘a’.

According to the definition of expected value, the above formula can berewritten as:

$\begin{matrix}{{E\left\lbrack \left( {\hat{a} - a} \right)^{2} \middle| x \right\rbrack} = {\int{\int{\left( {\hat{a} - a} \right)^{2}{\Pr\left( {a,x} \right)}{\mathbb{d}a}{\mathbb{d}x}}}}} \\{= {\int{\left\lbrack {\int{\left( {\hat{a} - a} \right)^{2}{\Pr\left( a \middle| x \right)}{\mathbb{d}a}}} \right\rbrack{\Pr(x)}{\mathbb{d}x}}}}\end{matrix}$

By partially differentiating the above formula, an equation below can bederived:

$\frac{\partial}{\partial\hat{a}}\begin{matrix}{{E\left\lbrack \left( {\hat{a} - a} \right)^{2} \middle| x \right\rbrack} = {\frac{\partial}{\partial\hat{a}}{\int{\left( {\hat{a} - a} \right)^{2}{\Pr\left( a \middle| x \right)}{\mathbb{d}a}}}}} \\{= {\int{\frac{\partial}{\partial\hat{a}}\left( {\hat{a} - a} \right)^{2}{\Pr\left( a \middle| x \right)}{\mathbb{d}a}}}} \\{= {\int{2\left( {\hat{a} - a} \right){\Pr\left( a \middle| x \right)}{\mathbb{d}a}}}} \\{= {{2\hat{a}{\int{{\Pr\left( a \middle| x \right)}{\mathbb{d}a}}}} - {2{\int{a\;{\Pr\left( a \middle| x \right)}{\mathbb{d}a}}}}}}\end{matrix}$

It is known from the above equations that, the optimized estimationresult â capable of minimizing the Bayesian risk R is a conditionalmean.â=∫aPr(a|x)da=E[a|x]

According to the definition of expected value, by respectivelymultiplying the candidate signals with a probability of occurrence, theabove formula can be rewritten as:

$\hat{a} = {{E\left\lbrack a \middle| x \right\rbrack} = {\sum\limits_{i \in \Omega}{s_{i}{\Pr\left( {a = \left. s_{i} \middle| x \right.} \right)}}}}$

By expanding the above formula based on the Bayesian theorem, a formulabelow is obtained:

$\begin{matrix}{\hat{a} = {\sum\limits_{i \in \Omega}{s_{i}\frac{{\Pr\left( x \middle| s_{i} \right)} \cdot {\Pr\left( s_{i} \right)}}{\Pr(x)}}}} \\{= {\sum\limits_{i \in \Omega}{s_{i}\frac{{\Pr\left( x \middle| s_{i} \right)}~ \cdot {\Pr\left( s_{i} \right)}}{\sum\limits_{j \in \Omega}{{\Pr\left( x \middle| s_{j} \right)} \cdot {\Pr\left( s_{j\;} \right)}}}}}}\end{matrix}$

Assuming that the probabilities of the original signal ‘a’ being equalto any of the candidate signals s_(i) are the same, the above formulacan be further simplified as:

$\hat{a} = {\sum\limits_{i \in \Omega}{s_{i}\frac{\Pr\left( x \middle| s_{i} \right)}{\sum\limits_{j \in \Omega}{\Pr\left( x \middle| s_{j} \right)}}}}$

Assuming that the foregoing phase-shift keying signal x is transmittedto the phase error correcting apparatus 100 through an AWGN channel, anda symbol σ represents a noise variance of the AWGN channel, the aboveformula can be rewritten as:

$\hat{a} = {\sum\limits_{i \in \Omega}{s_{i}\frac{{\mathbb{e}}^{- \frac{{{x - s_{i}}}^{2}}{2\sigma^{2}}}}{\sum\limits_{j \in \Omega}{\mathbb{e}}^{- \frac{{{x - s_{j}}}^{2}}{2\sigma^{2}}}}}}$

Assuming that an energy of the phase-shift keying signal x is a fixedvalue when the phase-shift keying signal x is inputted into the phaseerror correcting apparatus 100, the above formula can be rewritten as:

$\hat{a} = {\sum\limits_{i \in \Omega}{s_{i}\frac{{\mathbb{e}}^{\frac{{Re}{({x*s_{i}})}}{\sigma^{2}}}}{\sum\limits_{j \in \Omega}{\mathbb{e}}^{\frac{{Re}{({x*s_{j}})}}{\sigma^{2}}}}}}$

The phase difference between the phase-shift keying signal x and theestimation result â capable of minimizing the Bayesian risk R, i.e., thephase error θ between the phase-shift keying signal and the estimationresult a capable of minimizing the Bayesian risk R, is defined as:

$\theta = {{\arg\left\{ {x \cdot {\hat{a}}^{*}} \right\}} = {\tan^{- 1}\left\{ \frac{{Im}\left\{ {x \cdot {\hat{a}}^{*}} \right\}}{{Re}\left\{ {x \cdot {\hat{a}}^{*}} \right\}} \right\}}}$

By substituting the obtained estimation result a into the above, theabove formula can be rewritten as:

${\theta = {\tan^{- 1}\left\{ \frac{{Im}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{- \frac{{{x - s_{i}}}^{2}}{2\sigma^{2}}}}} \right\rbrack^{*}} \right\}}{{Re}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{- \frac{{{x - s_{i}}}^{2}}{2\sigma^{2}}}}} \right\rbrack^{*}} \right\}} \right\}}},{or}$$\theta = {\tan^{- 1}\left\{ \frac{{Im}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{\frac{{Re}{\{{x^{*}s_{i}}\}}}{\sigma^{2}}}}} \right\rbrack^{*}} \right\}}{{Re}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{\frac{{Re}{\{{x^{*}s_{i}}\}}}{\sigma^{2}}}}} \right\rbrack^{*}} \right\}} \right\}}$

In this embodiment, after receiving the phase-shift keying signal x, theerror estimating module 12 calculates the phase error θ according to thephase-shift keying signal x and the plurality of candidate signalss_(i), using one of the above two formulas. The correcting module 14then corrects the phase-shift keying signal x according to the phaseerror θ generated by the error estimating module 12. In practice, thecorrecting module 14 may be implemented by a phase derotator. Aspreviously stated, the estimation result a can render the minimizedBayesian risk R. Therefore, by correcting the phase-shift keying signalx to equal to or close to the estimation result a according to the phaseerror θ, the phase difference between the corrected phase-shift keyingsignal x′ and the original phase-shift keying signal a can be reduced orminimized.

In another embodiment, the error estimating module 12 may be designed toomit the arctan calculation in the above formulas to directly calculatethe phase error θ according to one of the two formulas below:

${\theta = \frac{{Im}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{- \frac{{{x - s_{i}}}^{2}}{2\sigma^{2}}}}} \right\rbrack^{*}} \right\}}{{Re}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{- \frac{{{x - s_{i}}}^{2}}{2\sigma^{2}}}}} \right\rbrack^{*}} \right\}}},{and}$$\theta = \frac{{Im}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{\frac{{Re}{\{{x^{*}s_{i}}\}}}{\sigma^{2}}}}} \right\rbrack^{*}} \right\}}{{Re}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{\frac{{Re}{\{{x^{*}s_{i}}\}}}{\sigma^{2}}}}} \right\rbrack^{*}} \right\}}$

When the phase error θ is relatively small (e.g., smaller than 5degrees), tan θ and θ are almost equal to each other, and so the errorestimating module 12 can still obtain a quite accurate result accordingto the simplified formulas. In other words, when the phase differencebetween the phase-shift keying signal x and the estimation result â isnot too large, the simplified formulas are feasible while hardwarecircuit complexities in the error estimating module 12 can be reduced.

FIG. 2A shows a phase error correcting apparatus 200 according toanother embodiment of the present invention. In addition to an errorestimating module 22 and a correcting module 24, the phase errorcorrecting apparatus 200 further comprises a low-pass filter 26 and anumerically-controlled oscillator (NCO) 28. In practice, the errorestimating module 22, the low-pass filter 26 and the NCO 28 may becombined to be a phase-locked loop (PLL). The low-pass filter 26 removesa high-frequency component from the phase error θ to generate a filteredresult. The NCO 28 subsequently generates an output signal according tothe filtered result to control the correcting module 24 to adjust thephase of the phase-shift keying signal x.

Before the error estimating module 22 performs phase error estimation onthe phase-shift keying signal x for the first time (i.e., before thecorrected phase-shift keying signal x′ is generated), the correctingmodule 24 may transmit the phase-shift keying signal x to the errorestimating module 22. It should be noted that, the error estimatingmodule 22 in this embodiment adopts the above simplified formulas tocalculate the phase error θ.

As previously described, given that the phase difference between thephase-shift keying signal x and the estimation result â is not toolarge, the phase error θ generated by the error estimating module 22equals the phase difference. In contrast, when the phase differencebetween the phase-shift keying signal x and the estimation result â israther large, the phase error θ generated by the error estimating module22 does not equal the phase difference. Theoretically, the positive andnegative signs of the phase error θ generated by the error estimatingmodule 22 are correct (i.e., the phase error θ and the phase differencehave the same positive/negative sign). After undergoing one or multiplecorrection, the corrected phase-shift keying signal x′ provided to theerror estimating module 22 to the correcting module 24 approaches theestimation result a capable of minimizing the Bayesian risk R.

Referring to FIG. 2B, the phase error correcting apparatus 200 furthercomprises a control module 27 for adjusting the low-pass filter 26. Ingeneral, phase errors come from thermal noise and phase noise. When thethermal noise is large, a response speed of the low-pass filter 26 canbe lowered to prevent the PLL from becoming unstable due to drasticphase changes. On the other hand, when the phase noise is large, theresponse speed of the low-pass filter 26 may be increased to catch upwith the phase change.

Therefore, in practice, the control module 27 may be designed toincrease the response time of the low-pass filter 26 when a thermalnoise index gets higher than a first threshold, and to increase theresponse speed of the low-pass filter when a phase noise index becomeshigher than a second threshold, thereby maintaining optimal systemperformance by adaptively adjusting the low-pass filter 26.

Referring to FIGS. 3A and 3B, the foregoing phase error correctingapparatuses 100 and 200 may further comprise a selecting module 30. Forexample, the selecting module 30 can be a hard slicer. According to thephase-shift keying signal x, the selecting module 30 selects a pluralityof candidate signals s_(i) that are closer to the phase-shift keyingsignal x from a plurality of original candidate signals, and providesthe selected candidate signals s_(i) to the error estimating module 22for calculating the phase error θ.

Taking an 8-PSK phase constellation in FIG. 4 as an example, if theselecting module 30 preliminarily determines that the phase of thephase-shift keying signal x falls within an interval corresponding to acandidate signal s₀, the selecting module 30 may suggest the errorestimating module 22 to consider only the candidate signal s₀ andneighboring signals s₁ and s₇ when calculating the phase error θ. Unlessa ratio of the noise in the channel is extremely high, the probabilitythat the original phase-shift keying signal a equals the candidatesignals s₂ and s₆ is very low, and so the candidate signals s₂ and s₆can then be eliminated. By reducing the number of the candidate signalss_(i), the procedure for calculating the phase error θ can be furthersimplified and accelerated.

It should be noted that the number of candidate signals provided to theerror estimating module 22 by the selecting module 30 is not limited tothree. In addition, even after the corrected phase-shift keying signalx′, of which phase is not entirely “locked”, is generated, the selectingmodule 30 may keep or select again the candidate signal s_(i) providedto the error estimating module 22 according to the corrected phase-shiftkeying signal x′.

FIG. 5 shows a flowchart of a signal processing method according toanother embodiment of the present invention. Referring to FIG. 5, inStep S51, a phase-shift keying signal is received. In Step S52, a phaseerror is calculated according to the phase-shift keying signal, aplurality of known candidate signals and Bayesian estimation. In StepS53, the phase-shift keying signal is corrected according to the phaseerror. Details of the signal processing method are as discussed inrelated descriptions of the phase error correcting apparatus 100 and 200in the foregoing embodiments, and shall be omitted herein.

Therefore, an apparatus and method for correcting a phase error isprovided by the embodiment of the present invention. In the phase errorcorrecting apparatus and method, Bayesian estimation is adopted forminimizing an error between an estimated phase result and a correctphase. Since Bayesian estimation is capable of providing optimized MAPperformance in an AWGN channel, the phase error correcting apparatus andmethod can accordingly generate a satisfactory phase estimation result.In addition, by selectively decreasing the number of candidate phases,costs and complexities of the phase error correcting apparatus andmethod can be further reduced.

It should be noted that, the phase error correcting apparatus of thepresent invention may also adopt other types of Bayesian cost functionsas calculation basis for estimating the phase error instead of using thequadratic cost function. In addition, the phase error correctingapparatus and method can be implemented in not only phase-shift keyingdigital communication systems but also other types of signal processingsystems with a demand of determining a signal phase error, so as toprovide satisfactory phase error determination.

While the invention has been described by way of example and in terms ofthe preferred embodiments, it is to be understood that the invention isnot limited thereto. On the contrary, it is intended to cover variousmodifications and similar arrangements and procedures, and the scope ofthe appended claims therefore should be accorded the broadestinterpretation so as to encompass all such modifications and similararrangements and procedures.

What is claimed is:
 1. A phase error correcting apparatus, applied to a phase-shift keying communication system, the apparatus comprising: an error estimating module, for receiving a phase-shift keying signal comprising an original signal and a noise signal, and calculating a phase error caused by the noise signal according to the phase-shift keying signal and a plurality of candidate signals using Bayesian estimation; wherein the original signal represents one of the plurality of candidate signals; and a correcting module, for correcting the phase-shift keying signal according to the phase error, wherein the error estimating module associates a quadratic of a difference of the original signal and an estimated signal corresponding to the phase-shift keying signal to a Bayesian risk, and calculates the phase error toward minimizing the Bayesian risk, in a way that a phase difference of the corrected phase-shift keying signal and the original signal is minimized.
 2. The apparatus according to claim 1, wherein the error estimating module calculates the phase error θ according to one of two formulas: ${\theta = {\tan^{- 1}\left\{ \frac{{Im}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{- \frac{{{x - s_{i}}}^{2}}{2\sigma^{2}}}}} \right\rbrack^{*}} \right\}}{{Re}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{- \frac{{{x - s_{i}}}^{2}}{2\sigma^{2}}}}} \right\rbrack^{*}} \right\}} \right\}}},{and}$ ${\theta = {\tan^{- 1}\left\{ \frac{{Im}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{\frac{{Re}{\{{x^{*}s_{i}}\}}}{\sigma^{2}}}}} \right\rbrack^{*}} \right\}}{{Re}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{\frac{{Re}{\{{x^{*}s_{i}}\}}}{\sigma^{2}}}}} \right\rbrack^{*}} \right\}} \right\}}};$ where x represents the phase-shifted keying signal, s_(i) represents the candidate signals, σ represents a noise variance of an additive white Gaussian noise (AWGN) channel.
 3. The apparatus according to claim 1, wherein the error estimating module calculates the phase error θ according to one of two formulas: ${\theta = \frac{{Im}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{- \frac{{{x - s_{i}}}^{2}}{2\sigma^{2}}}}} \right\rbrack^{*}} \right\}}{{Re}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{- \frac{{{x - s_{i}}}^{2}}{2\sigma^{2}}}}} \right\rbrack^{*}} \right\}}},{and}$ ${\theta = \frac{{Im}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{\frac{{Re}{\{{x^{*}s_{i}}\}}}{\sigma^{2}}}}} \right\rbrack^{*}} \right\}}{{Re}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{\frac{{Re}{\{{x^{*}s_{i}}\}}}{\sigma^{2}}}}} \right\rbrack^{*}} \right\}}};$ where x represents the phase-shifted keying signal, s_(i) represents the candidate signals, σ represents a noise variance of an AWGN channel.
 4. The apparatus according to claim 3, further comprising: a low-pass filter, for filtering the phase error to generate a filtered result; and a numerically-controlled oscillator (NCO), for generating an output signal according to the filtered result for the correcting module to adjust a phase of the phase-shift keying signal.
 5. The apparatus according to claim 4, further comprising: a control module, for adjusting the low-pass filter; wherein, when the control module determines a thermal noise index is higher than a first threshold, the control module lowers a response speed of the low-pass filter; and when the control module determines a phase noise index is higher than a second threshold, the control module raises the response speed of the low-pass filter.
 6. The apparatus according to claim 1, further comprising: a selecting module, for selecting the plurality of candidate signals for the error estimating module according to the phase-shift keying signal.
 7. A phase error correcting method, applicable to a phase-shift keying communication system, the method comprising: a) receiving a phase-shift keying signal comprising an original signal and a noise signal; b) calculating a phase error caused by the noise according to the phase-shift keying signal and a plurality of candidate signals using Bayesian estimation, wherein the original signal represents one of the candidate signals; and c) correcting the phase-shift keying signal according to the phase error, wherein step (b) comprises associating a quadratic of a difference of the original signal and an estimated signal corresponding to the phase-shift keying signal to a Bayesian risk, and calculating the phase error toward minimizing the Bayesian risk, in a way that a phase difference of the corrected phase-shift keying signal and the original signal is minimized.
 8. The method according to claim 7, wherein step (b) comprises calculating the phase error θ according to one of two formulas: ${\theta = {\tan^{- 1}\left\{ \frac{{Im}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{- \frac{{{x - s_{i}}}^{2}}{2\sigma^{2}}}}} \right\rbrack^{*}} \right\}}{{Re}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{- \frac{{{x - s_{i}}}^{2}}{2\sigma^{2}}}}} \right\rbrack^{*}} \right\}} \right\}}},{and}$ ${\theta = {\tan^{- 1}\left\{ \frac{{Im}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{\frac{{Re}{\{{x^{*}s_{i}}\}}}{\sigma^{2}}}}} \right\rbrack^{*}} \right\}}{{Re}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{\frac{{Re}{\{{x^{*}s_{i}}\}}}{\sigma^{2}}}}} \right\rbrack^{*}} \right\}} \right\}}};$ where x represents the phase-shifted keying signal, s_(i) represents the candidate signals, σ represents a noise variance of an AWGN channel.
 9. The method according to claim 7, wherein step (b) comprises calculating the phase error θ according to one of two formulas: ${\theta = \frac{{Im}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{- \frac{{{x - s_{i}}}^{2}}{2\sigma^{2}}}}} \right\rbrack^{*}} \right\}}{{Re}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{- \frac{{{x - s_{i}}}^{2}}{2\sigma^{2}}}}} \right\rbrack^{*}} \right\}}},{and}$ ${\theta = \frac{{Im}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{\frac{{Re}{\{{x^{*}s_{i}}\}}}{\sigma^{2}}}}} \right\rbrack^{*}} \right\}}{{Re}\left\{ {x \cdot \left\lbrack {\sum\limits_{i \in \Omega}{s_{i} \cdot {\mathbb{e}}^{\frac{{Re}{\{{x^{*}s_{i}}\}}}{\sigma^{2}}}}} \right\rbrack^{*}} \right\}}};$ where x represents the phase-shifted keying signal, s_(i) represents the candidate signals, σ represents a noise variance of an AWGN channel.
 10. The method according to claim 9, wherein step (c) comprises: low-pass filtering the phase error to generate a filtered result; and generating an output signal according to the filtered result for adjusting a phase of the phase-shift keying signal.
 11. The method according to claim 10, further comprising: when determining that a thermal noise index is higher than a first threshold, lowering a response speed of the low-pass filter; and when determining that a phase noise index is higher than a second threshold, raising the response speed of the low-pass filter.
 12. The method according to claim 7, further comprising: selecting the plurality of candidate signals according to the phase-shift keying signal for use of step (b). 